21.10. SOME EMBEDDING THEOREMS 703
The idea is to show that un approximates un well and then to argue that a subsequence ofthe {un} is a Cauchy sequence yielding a contradiction to 21.10.51.
Therefore,
un (t)−un (t) =k
∑i=1
un (t)X[ti−1,ti) (t)−k
∑i=1
uniX[ti−1,ti) (t)
=k
∑i=1
1ti− ti−1
∫ ti
ti−1
un (t)dsX[ti−1,ti) (t)−k
∑i=1
1ti− ti−1
∫ ti
ti−1
un (s)dsX[ti−1,ti) (t)
=k
∑i=1
1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))dsX[ti−1,ti) (t) .
It follows from Jensen’s inequality that
||un (t)−un (t)||pW
=k
∑i=1
∣∣∣∣∣∣∣∣ 1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))ds∣∣∣∣∣∣∣∣p
WX[ti−1,ti) (t)
And so ∫ T
0||un (t)−un (t)||pW dt =
k
∑i=1
∫ ti
ti−1
∣∣∣∣∣∣∣∣ 1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))ds∣∣∣∣∣∣∣∣p
Wdt
≤k
∑i=1
∫ ti
ti−1
ε
∥∥∥∥ 1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))ds∥∥∥∥p
Edt
+Cε
k
∑i=1
∫ ti
ti−1
∥∥∥∥ 1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))ds∥∥∥∥p
Xdt (21.10.52)
Consider the second of these. It equals
Cε
k
∑i=1
∫ ti
ti−1
∥∥∥∥ 1ti− ti−1
∫ ti
ti−1
∫ t
su′n (τ)dτds
∥∥∥∥p
Xdt
This is no larger than
≤Cε
k
∑i=1
∫ ti
ti−1
(1
ti− ti−1
∫ ti
ti−1
∫ ti
ti−1
∥∥u′n (τ)∥∥
X dτds)p
dt
=Cε
k
∑i=1
∫ ti
ti−1
(∫ ti
ti−1
∥∥u′n (τ)∥∥
X dτ
)p
dt
=Cε
k
∑i=1
((ti− ti−1)
1/p∫ ti
ti−1
∥∥u′n (τ)∥∥
X dτ
)p